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Bohr model — energy levels

Calculate the energy of level n=3 in hydrogen (En = -13.6/n² eV) and the n=3 → n=2 transition energy.
E2 = -3.40 eV (given).
Given data
E_n = -13.6/n² eVE₂ = -3.40 eVhc = 1240 eV·nm
Review the theory: Struttura Elettronica
Steps
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  1. Calculate the energy of level n=3.
  2. Calculate ΔE for the n=3 → n=2 transition.
  3. Calculate the wavelength of the emitted photon (λ = hc/ΔE).
Full worked solution
  1. Calculate the energy of level n=3.
    E3=−13.632E_3 = -\frac{13.6}{3^2}
    E3=−13.6/32=−13.6/9=−1.511 eVE_3 = -13.6/3^2 = -13.6/9 = \mathbf{-1.511\,eV}. The energy of level n=3 is obtained by substituting n=3 into the Bohr formula.
  2. Calculate ΔE for the n=3 → n=2 transition.
    ΔE=E3−E2\Delta E = E_3 - E_2
    ΔE=E3−E2=−1.511−(−3.40)=1.889 eV\Delta E = E_3 - E_2 = -1.511 - (-3.40) = \mathbf{1.889\,eV}. The energy difference between levels n=3 and n=2 is 1.889 eV.
  3. Calculate the wavelength of the emitted photon (λ = hc/ΔE).
    λ=12401.889\lambda = \frac{1240}{1.889}
    λ=hc/ΔE=1240/1.889=656 nm\lambda = hc/\Delta E = 1240/1.889 = \mathbf{656\,nm}. The wavelength of the emitted photon corresponds to the Balmer red line.
Result:E₃ = -1.511 eV, ΔE = 1.889 eV, λ = 656 nm.
TheoryStruttura ElettronicaFormula sheetChemistry

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